The recent discovery of room-temperature superconductivity does not represent the
end of research in superconductivity. Starting with the first discovery of superconductivity
in 1911, the study of superconducting properties progressed from phenomenology
to the microscopic perspective BCS theory. However, with the discovery of a series
of unconventional superconductors, the progress of related theories encountered significant
difficulties. A possible path to understanding unconventional superconductivity
is to distinguish three-dimensional, (quasi-)two-dimensional, and (quasi-) one-dimensional
superconductors to investigate the variation of superconducting properties
as a function of dimensionality. In this thesis, we study a series of superconductors
in different dimensionalities to uncover the properties of unconventional superconductors.

In Chapter 3, we investigate the interfacial superconductivity generated by the
proximity effect from a superconducting niobium layer deposited onto a quantum
anomalous Hall insulator (QAHI) (Cr

_{0.12}Bi

_{0.26}Sb

_{0.62})

_{2}Te

_{3} through point contact spectroscopy
measurements. By varying the magnitude and sign of the applied magnetic
field in the perpendicular direction to the film, we can study the proximity-induced
superconductivity when the QAHI magnetization is reversed. According to the theoretical
prediction, the superconductivity formed in the QAHI boundary state is a
chiral topological superconducting state, and its quasiparticles are chiral Majorana
modes. Due to the local nature of the point-contact measurements, we probe the superconducting
spectra on a length scale smaller than the size of magnetic domains, thus avoiding the effects of a chaotic and stochastically distributed nature. The topological
properties of the QAHI/Nb heterostructures change abruptly during the magnetization
reversal when the Chern number changes from N=±2 (represented by the state
with two chiral Majorana modes) to N=±1 (one single Majorana mode) and passing
through a trivial insulating state with N=0. Our measurements on three devices indicate
this series of topological transitions, which agree with the theoretical predictions
of a dip-like feature in the point-contact spectra for the N=±2 state and a plateau-like
feature for N=±1 and represents the existence of chiral Majorana modes.

In Chapter 4, we investigate the helical conductance in quasi-1D nanoribbons of
the same (Cr

_{0.12}Bi

_{0.26}Sb

_{0.62})

_{2}Te

_{3} quantum anomalous Hall insulator (QAHI). We fabricate
the nanoribbons with a width between 75 and 100nm with the help of gallium focused
ion beam system. When shrinking the width of the quantum anomalous Hall insulator,
the chiral edge modes from opposite sides should hybridize at a critical width
of ~ 100nm, leading to the opening of a hybridization gap and a single helical conduction
channel across the nanoribbon. Our magnetotransport experiment demonstrates
that the chiral edge mode of the QAHI can be channeled through such narrow nanoribbons
without significant dissipation, thus suggesting that a helical transport channel
is indeed formed.

Chapter 5 investigates the superconducting gap symmetry of NbSe

_{2} from monolayer
to few-layer thickness by applying a magnetic field, which is rotated in the plane.
2D NbSe

_{2} exhibits a special type of Ising spin-orbit coupling, which makes firmly pins
the electron spins to the out-of-plane direction. The Ising SOC causes a so-called Ising
superconducting state, which helps the superconductor withstand very high in-plane
magnetic fields that far exceed the Pauli limit for superconductivity. Detailed measurements
were performed on several samples, and it was found that for the monolayers, a
sixfold nodal symmetry of the upper critical field appears, which agrees perfectly with
the theoretical prediction of a nodal topological superconducting phase with six pairs
of point nodes that should exist near the upper critical field in high parallel fields. Surprisingly,
at lower fields, the resistivity exhibits a twofold in-plane symmetry, which is
at odds with the trifold crystalline symmetry, similar to the nematic order previously
observed in Nb-doped Bi

_{2}Se

_{3}. We attribute the latter’s origin to the presence of several
competing superconducting channels.

Chapter 6 extends our investigation to bulk transition metal dichalcogenide superconductors
in high parallel magnetic fields to investigate whether Ising superconductivity
or other unusual superconducting states can also exist in bulk. By strictly
controlling the alignment of the magnetic field parallel to the layered structure while
measuring magnetic torque, specific heat, and thermal expansion, we find that the
upper critical magnetic field in the NbS

_{2} bulk single crystals (a material similar to
NbSe

_{2}) also exceeds the Pauli limit. The upper critical field first shows a saturation
near the Pauli limit of 10 T in the H-T phase diagram, followed by a pronounced upturn
at lower temperatures. In addition, a phase transition anomaly is observed near
10T within the superconducting state. These observations agree with the formation of
a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. Our theoretical calculations show
that Ising superconductivity does not play a major role. In the bulk case, the Fulde-
Ferrell-Larkin-Ovchinnikov state is responsible for the superconductivity in magnetic
fields above the Pauli limit for superconductivity.

Finally, by studying thinner samples of NbS

_{2}, thus inducing the crossover to a 2D
material, we investigate the effect of anisotropy as a function of thickness. We prepared
and studied exfoliated NbS

_{2} thin films with thicknesses in the crossover region
between 2D materials and the bulk. Combining in-plane and out-of-plane angular-dependent
electric magnetotransport measurements, we find that the materials show
quasi-2D superconducting properties, possibly due to the weak coupling between the
layers.

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